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arXiv:0903.2432 · Physical Review B

Operator space entanglement entropy in X Y spin chains

Iztok Pižorn, Tomaž Prosen

Published May 13, 2009• 103 citations
arXivHTMLPDF

Abstract

The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy (OSEE). We show that in the homogeneous Heisenberg $XY$ spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time. The prefactor in front of the logarithm generally depends only on the number of stationary points of the quasiparticle dispersion relation and for the $XY$ model changes from 1/3 to 2/3 exactly at the point of quantum phase transition to long-range magnetic correlations in the nonequilibrium steady state. In addition, we show that the presence of a small disorder triggers a saturation of the OSEE.

DOI
10.1103/physrevb.79.184416

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The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy (OSEE). We show that in the homogeneous Heisenberg XYXYXY spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time. The prefactor in front of the logarithm generally depends only on the number of stationary points of the quasiparticle dispersion relation and for the XYXYXY model changes from 1/3 to 2/3 exactly at the point of quantum phase transition to long-range magnetic correlations in the nonequilibrium steady state. In addition, we show that the presence of a small disorder triggers a saturation of the OSEE.
Understanding the spreading of the operator space entanglement entropy (OSEE) is key in order to explore out-of-equilibrium quantum many-body systems. Here we argue that for integrable models the dynamics of the OSEE is related to the diffusion of the operator front. We derive the logarithmic bound 1/2ln(t)1/2ln(t)1/2ln(t) for the OSEE of some simple, i.e., low-rank, diagonal local operators. We numerically check that the bound is saturated in the rule 54 chain, which is representative of interacting integrable systems. Remarkably, the same bound is saturated in the spin-1/2 Heisenberg XXZXXZXXZ chain. Away from the isotropic point and from the free-fermion point, the OSEE grows as 1/2ln(t)1/2ln(t)1/2ln(t), irrespective of the chain anisotropy, suggesting universality. Finally, we discuss the effect of integrability breaking. We show that strong finite-time effects are present, which prevent us from probing the asymptotic behavior of the OSEE.
We study the von Neumann operator-space entanglement entropy (OSEE) of initial product operators evolved under random gaussian dynamics, as modeled by a Brownian SYK2_22​ model. By exploiting a mapping to a domain-wall melting problem, we derive an exact expression for the time-dependent bipartite OSEE in the thermodynamic limit, and obtain an analytic characterization of the corresponding gaussian Page curve emerging at late times. Our results build on recent work on the entanglement dynamics in the all-to-all quantum simple exclusion process, leveraging tools from RMT, including the Coulomb-gas approach and a mapping to the free Jacobi process. Finally, based on our results, we discuss the possibility of relating the OSEE with non-gaussianity measures in the Heisenberg picture.
The efficiency of time-dependent density matrix renormalization group methods is intrinsically connected to the rate of entanglement growth. We introduce a measure of entanglement in the space of operators and show, for a transverse Ising spin-1∕21∕21∕2 chain, that the simulation of observables, contrary to the simulation of typical pure quantum states, is efficient for initial local operators. For initial operators with a finite index in Majorana representation, the operator space entanglement entropy saturates with time to a level which is calculated analytically, while for initial operators with infinite index the growth of operator space entanglement entropy is shown to be logarithmic.
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We explore the long-time behavior of Local Operator Entanglement entropy (LOE) in finite-size interacting integrable systems. For certain operators in the Rule 54 automaton, we prove that the LOE saturates to a value that is at most logarithmic in system size. This bound extends previous work [Phys. Rev. Lett. 122, 250603 (2019); Commun. Math. Phys. 371, 651 (2019)] showing LOE grows logarithmically in the early time regime, t\ll L t ≪ L , to the late time regime, $tL $. However, the late-time logarithmic bound relies on a feature of Rule 54 that does not generalize to other interacting integrable systems: namely, that there are only two types of quasiparticles, and therefore only two possible values of the phase shift between quasiparticles. We present a heuristic argument, supported by numerical evidence, that for generic interacting integrable systems (such as the Heisenberg spin chain) the saturated value of the LOE is volume-law in system size.
Free fermion models with an operator entanglement growing as O(t)O(\sqrt{t})O(t​)
We study the time evolution of the Operator Space Entanglement Entropy (OSEE) of one-dimensional number-conserving free fermions subject to a random space- and time-dependent evolution. It is known that, in free-fermion systems without randomness, the OSEE of \emph{local} operators grows at most \emph{logarithmically} in time. Here we show that, in the presence of randomness, the OSEE of local spin operators involving a Jordan-Wigner string grows as O(t)\mathcal O(\sqrt t)O(t​), for all values of the Rényi index. We establish this in two models: a Brownian Hamiltonian and a brick-wall matchgate circuit. We also consider the OSEE of the evolution operator, and we show that it also grows as O(t)\mathcal O(\sqrt{t})O(t​), in contrast with the linear growth generically expected without randomness. We analytically compute the average entropies of the two operators by mapping the quantum models to the symmetric simple exclusion process, relying on an assumption of self-averaging which we support numerically. We also consider two other types of randomness: random evolution in time that keeps spatial translation invariance, and quenched disorder. In the former case, we find that the OSEE also grows as t\sqrt{t}t​. In the latter case, we find that the entropy grows as O(log⁡log⁡t)\mathcal O( \log \log t)O(loglogt) for Hamiltonian evolution both for local operators with a Jordan-Wigner string and for the evolution operator, whereas it remains bounded at all times in the brick-wall matchgate circuit.